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Elsevier, Journal of Mathematical Analysis and Applications, 2(361), p. 533-542

DOI: 10.1016/j.jmaa.2009.07.034

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Asymptotic behaviour for small mass in the two-dimensional parabolic–elliptic Keller–Segel model

Journal article published in 2010 by Adrien Blanchet, Jean Dolbeault ORCID, Miguel Escobedo, Javier Fernández
This paper is made freely available by the publisher.
This paper is made freely available by the publisher.

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Abstract

The Keller-Segel system describes the collective motion of cells that are attracted by a chemical substance and are able to emit it. In its simplest form, it is a conservative drift-diffusion equation for the cell density coupled to an elliptic equation for the chemo-attractant concentration. This paper deals with the rate of convergence towards a unique stationary state in self-similar variables, which describes the intermediate asymptotics of the solutions in the original variables. Although it is known that solutions globally exist for any mass less $8π\,$, a smaller mass condition is needed in our approach for proving an exponential rate of convergence in self-similar~variables.